Segmentation of Multivariate Mixed Data via Lossy Data Coding and Compression

Abstract

In this paper, based on ideas from lossy data coding and compression, we present a simple but effective technique for segmenting multivariate mixed data that are drawn from a mixture of Gaussian distributions, which are allowed to be almost degenerate. The goal is to find the optimal segmentation that minimizes the overall coding length of the segmented data, subject to a given distortion. By analyzing the coding length/rate of mixed data, we formally establish some strong connections of data segmentation to many fundamental concepts in lossy data compression and rate distortion theory. We show that a deterministic segmentation is approximately the (asymptotically) optimal solution for compressing mixed data. We propose a very simple and effective algorithm which depends on a single parameter, the allowable distortion. At any given distortion, the algorithm automatically determines the corresponding number and dimension of the groups and does not involve any parameter estimation. Simulation results reveal intriguing phase-transition-like behaviors of the number of segments when changing the level of distortion or the amount of outliers. Finally, we demonstrate how this technique can be readily applied to segment real imagery and bioinformatic data.

Visual Saliency Based on Conditional Entropy

“…Note that the first term of the equation stands for the coding length required to codeW , and the second term of the equation stands for the additional coding length of the mean vector. The equation also gives a good upper bound for degenerated Gaussian data or subspace-like data [15]. Moreover, the coding length has proven to be effective for clustering [16] and classification [15].…”

“…The equation also gives a good upper bound for degenerated Gaussian data or subspace-like data [15]. Moreover, the coding length has proven to be effective for clustering [16] and classification [15].…”

“…We first present a brief introduction to the lossy coding length of multivariate Gaussian data [15], which is used to calculate the visual saliency in the following section. Assume a set of vectors W = (w 1 , w 2 , ..., w m ) ∈ R n×m is presented, a lossy coding scheme L(·) maps W to a sequence of binary bits.…”

“…Although such a simplified assumption is hardly satisfied in real situations, experimental results indicate a robust solution to the visual saliency. In fact, the assumption can be extended to degenerated Gaussian or sub space like data [15].…”

A human visual system-driven image segmentation algorithm

“…The segmented images are expected to consist of regions within which the image content is homogeneous, while the contrast between neighboring regions is high. Typical methods falling into this category include region growing, watershed, some MRF-based methods [3], mean-shift [9] and the recently presented lossy data compression-based approach [10]. Segmentation methods based on the boundary or edge information are designed to exploit the discontinuity of the image features, such as difference in texture or pixel intensity, on the two sides of the boundary.…”